Discuss the continuity of f(x)= |x|

Question

Discuss the continuity of $\text{f(x)}=\sin|\text{x}|$

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Answer

Let $\text{f(x)}=\sin|\text{x}|$ This function f is defined for every real number and f can be written as the composition of two functions as, f = goh, where g(x) = |x| and $\text{h(x)}=\sin\text{x}$ $\big[\because (\text{goh})(\text{x})=\text{g(h(x))}=\text{g}(\sin\text{x})=|\sin\text{x}|=\text{f(x)}]\Big]$ It has to be proved first that g(x) = |x| and $\text{h(x)}=\sin\text{x}$ are continuous functions. g(x) = |x| can be written as $\text{g(x)}=\begin{cases}-\text{x},&\text{if }\text{ x}<0\\\text{x},&\text{if }\text{ x}\geq0\end{cases}$ Clearly, g is defined for all real numbers. Let c be real number.Case I:
If c < 0, then g(c) = -c and $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}(-\text{x})=-\text{c}$ $\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\text{g(c)}$ Therefore, g is continuous at all points x, such that x < 0Case II:
If c > 0, then g(c) = c and $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}(\text{x})=\text{c}$ $\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\text{g(c)}$ Therefore, g is continuous at all points x, such that x > 0Case III:
If c = 0, then g(c) = g(0) = 0 $\lim\limits_{\text{x}\rightarrow0^-}\text{g(x)}=\lim\limits_{\text{x}\rightarrow0^-}\text{g}(-\text{x})=0$ $\lim\limits_{\text{x}\rightarrow0^+}\text{g(x)}=\lim\limits_{\text{x}\rightarrow0^+}\text{g(x)}=0$ $\therefore\ \lim\limits_{\text{x}\rightarrow0^-}\text{g(x)}=\lim\limits_{\text{x}\rightarrow0^+}\text{g(x)}=\text{g}(0)$ Therefore, g is continuous at x = 0 From the above three observations, it can be concluded that g is continuous at all points. $\text{h(x)}=\sin\text{x}$ It is evident that $\text{h(x)}=\sin\text{x}$ is defined for every real number. Let c be a real number. Put x = c + k If x → c, then k → 0 $\text{h(c)}=\sin\text{c}$ $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{h(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}\sin\text{x}$ $=\lim\limits_{{\text{x}}\rightarrow\text{c}}\sin(\text{c}+\text{k})$ $=\lim\limits_{{\text{x}}\rightarrow\text{c}}\big[\sin\text{c}\cos\text{k}+\cos\text{c}\sin\text{k}\big]$ $=\lim\limits_{{\text{x}}\rightarrow\text{c}}(\sin\text{c}\cos\text{k})+\lim\limits_{{\text{x}}\rightarrow\text{c}}(\cos\text{c}\sin\text{k})$ $=\sin\text{c}\cos0+\cos\text{c}\sin0$ $=\sin\text{c}+0$ $=\sin\text{c}$ $\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{h(x)}=\text{g(c)}$ Therefore, h is a continuous function. It is know that for real valued functions g and h, such that (goh) is defind at c, if g is continuse at c and if f is continuous at g(c), then (fog) is continuous at c. Therefore, $\text{f(x)}=(\text{goh})(\text{x})=\text{g(h(x))}=\text{g}(\sin\text{x})=|\sin\text{x}|$ is a continuous function.